I know I'm resurrecting an ancient thread, but I was looking at this again while thinking about my Expand.pm module: I think the counts you showed were off: of the 2**64 = 18e18 combinations of the 64bit value, you listed 18e15 non-normalized numbers, of which 9e15 were denormalized, 18e18 were signaling NaN, and 18e18 were quiet NaN, which didn't quite add up right. :-) Here are my tallies:
Zeroes plus denormals are all the combinations that have the 11 exponent bits of 000_0000_0000, that is:
*_000_0000_0000_****_****_****_****_****_****_****_****_****_****_
+****_****_****
=> 2^53 = 9,007,199,254,740,992
=> #zeroes = 2
=> #denormals = 9,007,199,254,740,990
=> #positive = 4,503,599,627,370,495
=> #negative = 4,503,599,627,370,495
Infinities plus NaNs take all combinations that have the 11 exponent bits of 111_1111_1111, that is:
*_000_0000_0000_****_****_****_****_****_****_****_****_****_****_
+****_****_****
=> 2^53 = 9,007,199,254,740,992
=> #infinities = 2
=> #NaNs= 9,007,199,254,740,990
=> #signalling = 4,503,599,627,370,495
=> #quiet = 4,503,599,627,370,495
That should leave 2**64 - 2*(2**53) for normalized numbers:
18,446,744,073,709,551,616 = #64bit values
- 18,014,398,509,481,984 = #zeroes + #infinities + #denormal + #
+NaN
= 18,428,729,675,200,069,632 = #normalized numbers
Confirming 2**52, 2**53, 2**54, 2**64, and 2**64 - 2**54 :
cmd.exe> perl -MMath::BigInt -E "$b = Math::BigInt->new(2); my $x = ($
+b**$ARGV[0]); $x =~ s/(\d{1,3}?)(?=(\d{3})+$)/$1,/g ; say $x" 52
4,503,599,627,370,496
cmd.exe> perl -MMath::BigInt -E "$b = Math::BigInt->new(2); my $x = ($
+b**$ARGV[0]); $x =~ s/(\d{1,3}?)(?=(\d{3})+$)/$1,/g ; say $x" 53
9,007,199,254,740,992
cmd.exe> perl -MMath::BigInt -E "$b = Math::BigInt->new(2); my $x = ($
+b**$ARGV[0]); $x =~ s/(\d{1,3}?)(?=(\d{3})+$)/$1,/g ; say $x" 54
18,014,398,509,481,984
cmd.exe> perl -MMath::BigInt -E "$b = Math::BigInt->new(2); my $x = ($
+b**$ARGV[0]); $x =~ s/(\d{1,3}?)(?=(\d{3})+$)/$1,/g ; say $x" 64
18,446,744,073,709,551,616
cmd.exe> perl -MMath::BigInt -E "$b = Math::BigInt->new(2); $p1 = Math
+::BigInt->new($ARGV[0]); $p2 = Math::BigInt->new($ARGV[1]); $x = $b**
+$p1; $y = $b**$p2; $z = $x-$y; say join qq|\n|, map { s/(\d{1,3}?)(?=
+(\d{3})+$)/$1,/g; $_ } $x, $y, $z;" 64 54
18,446,744,073,709,551,616
18,014,398,509,481,984
18,428,729,675,200,069,632
